Homotopy Coherent Structures
نویسنده
چکیده
Naturally occurring diagrams in algebraic topology are commutative up to homotopy, but not on the nose. It was quickly realized that very little can be done with this information. Homotopy coherent category theory arose out of a desire to catalog the higher homotopical information required to restore constructibility (or more precisely, functoriality) in such “up to homotopy” settings. The first lecture will survey the classical theory of homotopy coherent diagrams of topological spaces. The second lecture will revisit the free resolutions used to define homotopy coherent diagrams and prove that they can also be understood as homotopy coherent realizations. This explains why diagrams valued in homotopy coherent nerves or more general ∞-categories are automatically homotopy coherent. The final lecture will venture into homotopy coherent algebra, connecting the newly discovered notion of homotopy coherent adjunction to the classical cobar and bar resolutions for homotopy coherent algebras.
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